I want to create a problem in which one needs to compute the sum of the numbers in a column of the Pascal's triangle i.e.
$$ {3 \choose 3 } + {4 \choose 3 } + \cdots + {10 \choose 3 }$$
where the person is supposed to know that this sum equals ${11 \choose 4 }$
This sum has to come up in a "natural" way. For instance, if the problem were like this, then you would need the product of products of three consecutive numbers:
"In 1950, three people participate in a copetition. In 1951, four people participate it. In 1952, five people participate it. And so on until 2022. What is the number of different combinations of podiums?"
(Note that although I suggested creating a natural problem, this example I showed is kind of forced and unnatural. I just want something not too forced like straight up asking the sum)
If this was the question, then the answer would be $ (1 \cdot 2 \cdot 3) \cdot (2 \cdot 3 \cdot 4) \cdot \dots \cdot (73 \cdot 74 \cdot 75)$. But what I want is some situation where the problem requires you to compute $ 1\cdot 2 \cdot 3 + \dots + 73 \cdot 74 \cdot 75$, which, in turn, would be much easier to solve if the pascal's triangle identity were known.
I also considerer asking the sum of the number of different podiums in each competition too forced (although the competition getting one competitor each year is also a little bit forced, but still acceptable for me).
Example Problem 1
How many North-East lattice paths exist from $(0,0)$ to $(3,y)$ where $0\leq y \leq 7$?
NE lattice path is a well-known problem and its solution is also widely known to utilise combination, for this particular problem the number of paths is given by the following expression:
$$ \sum_{y=0}^{7}\binom{3+y}{3} = \binom{3}{3}+\binom{4}{3}+...+\binom{10}{3} $$
Example Problem 2
How many ways are there to distribute undistinguishable candies to four children such that each child receive at least one but the total number of candies cannot exceed 11?
Stars and bars is among the most famous combinatorics problem. The number of ways to distribute the candies are given by the following expression:
$$ \sum_{n=4}^{11}\binom{n-1}{4-1}= \binom{3}{3}+\binom{4}{3}+...+\binom{10}{3} $$
Example Problem 3
$10$ children joined a competition and each ranks from first to tenth at the end. At least three of the top ranks receive flower. Three of the flowers contain money. How many possibilities exist?
Not as well known as the other two. Rank $1$ to $m$ receive flowers where $3\leq m\leq 10$ and three of these $m$ flowers contain money. Number of possibilities are given by the following expression.
$$ \sum_{m=3}^{10}\binom{m}{3} = \binom{3}{3}+\binom{4}{3}+...+\binom{10}{3} $$
And many more...